Properties

Label 2-2671-1.1-c1-0-86
Degree $2$
Conductor $2671$
Sign $-1$
Analytic cond. $21.3280$
Root an. cond. $4.61822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.34·2-s − 1.17·3-s + 3.51·4-s + 0.325·5-s + 2.75·6-s − 3.11·7-s − 3.55·8-s − 1.62·9-s − 0.764·10-s + 0.901·11-s − 4.12·12-s + 0.278·13-s + 7.30·14-s − 0.382·15-s + 1.31·16-s + 4.78·17-s + 3.81·18-s − 6.54·19-s + 1.14·20-s + 3.65·21-s − 2.11·22-s + 0.624·23-s + 4.17·24-s − 4.89·25-s − 0.654·26-s + 5.42·27-s − 10.9·28-s + ⋯
L(s)  = 1  − 1.66·2-s − 0.677·3-s + 1.75·4-s + 0.145·5-s + 1.12·6-s − 1.17·7-s − 1.25·8-s − 0.540·9-s − 0.241·10-s + 0.271·11-s − 1.19·12-s + 0.0773·13-s + 1.95·14-s − 0.0986·15-s + 0.329·16-s + 1.15·17-s + 0.898·18-s − 1.50·19-s + 0.255·20-s + 0.796·21-s − 0.451·22-s + 0.130·23-s + 0.851·24-s − 0.978·25-s − 0.128·26-s + 1.04·27-s − 2.06·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2671\)
Sign: $-1$
Analytic conductor: \(21.3280\)
Root analytic conductor: \(4.61822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2671} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2671,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2671 \( 1 + T \)
good2 \( 1 + 2.34T + 2T^{2} \)
3 \( 1 + 1.17T + 3T^{2} \)
5 \( 1 - 0.325T + 5T^{2} \)
7 \( 1 + 3.11T + 7T^{2} \)
11 \( 1 - 0.901T + 11T^{2} \)
13 \( 1 - 0.278T + 13T^{2} \)
17 \( 1 - 4.78T + 17T^{2} \)
19 \( 1 + 6.54T + 19T^{2} \)
23 \( 1 - 0.624T + 23T^{2} \)
29 \( 1 - 4.53T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 2.02T + 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 - 7.56T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 1.60T + 53T^{2} \)
59 \( 1 - 8.21T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 3.12T + 71T^{2} \)
73 \( 1 + 2.36T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 + 4.77T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.428115460409275712969152698777, −8.050053496747146730081292551951, −6.88098480352143738880266730904, −6.37485421083771572990850935128, −5.86086297379818688685920332298, −4.57717106055686410975937375279, −3.27095999869493584434065467572, −2.36379696874767080579924053372, −1.01235818421709302264576185758, 0, 1.01235818421709302264576185758, 2.36379696874767080579924053372, 3.27095999869493584434065467572, 4.57717106055686410975937375279, 5.86086297379818688685920332298, 6.37485421083771572990850935128, 6.88098480352143738880266730904, 8.050053496747146730081292551951, 8.428115460409275712969152698777

Graph of the $Z$-function along the critical line